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Left Equalizer Simple Semigroups

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Left Equalizer Simple Semigroups LEFT EQUALIZER SIMPLE SEMIGROUPS 1 ATTILA NAGY Department of Algebra, Mathematical Institute Budapest University of Technology and Economics 1521 Budapest, PO Box 91 e-mail: [email protected] Abstract In this paper we characterize and construct semigroups whose right regular representation is a left cancellative semigroup. These semigroups will be called left equalizer simple semigroups. For a congruence ̺ on a semigroup S, let F[̺] denote the ideal of the semigroup algebra F[S] which determines the kernel of the extended homomorphism of F[S] onto F[S/̺] induced by the canonical homomorphism of S onto S/̺. We examine the right colons (F[̺] :r F[S]) = {a ∈ F[S] : F[S]a ⊆ F[̺]} in general, and in that special case when ̺ has the property that the factor semigroup S/̺ is left equalizer simple. 1 Introduction and motivation Let S be a semigroup. For an arbitrary element a of S, let ̺a denote the transformation of S defined by ̺a : s 7→ sa (s ∈ S). It is well known that θS = {(a,b) ∈ S × S : (∀x ∈ S) xa = xb} is a congruence on S; this congruence is the kernel of the homomorphism ϕ : a 7→ ̺a of S into the semigroup of all right translations of S. The homomorphism ϕ is called the right regular representation of S; this is faithful if and only if S is a left reductive semigroup (that is, whenever xa = xb for some a,b ∈ S and for all x ∈ S then a = b). For convenience (as in [3] or [4]), the semigroup ϕ(S) is also called the right regular representation of S. It is an interesting problem to find couples (C 1, C2) of classes C1 and C2 of semigroups for which the next assertion is true. A semigroup S belongs to C1 if and only if the right regular representation of S belongs to C2. It is easy to see that a semigroup S is right commutative (that is, it satisfies the identity xab = xba ([5])) if and only if the right regular representation of S arXiv:1504.07183v2 [math.GR] 29 Sep 2015 is a commutative semigroup. A semigroup S is medial (that is, it satisfies the identity xaby = xbay ([5])) if and only if the right regular representation of S is a left commutative semigroup (that is, it satisfies the identity abx = bax ([5])). In [11], a semigroup S is called an M-inversive semigroup if, for each a ∈ S, there are elements x, y ∈ S such that ax and ya are middle units of S, that is, caxd = cd and cyad = cd is satisfied for all c, d ∈ S. In [4], it is proved that a semigroup S is M-inversive if and only if the right regular representation of S is a right group (that is, a direct product of a right zero semigroup (satisfying the identity ab = b) and a group). 1Keywords: semigroup, congruence, semigroup algebra. MSC(2010): 20M10, 20M25. The paper will be published in Acta Mathematica Hungarica 1 In Section 2, we define a special class of semigroups which contains all of the M-inversive ones. The semigroups belonging to this class will be called left equalizer simple semigroups. Extending the above mentioned result of [4], we show that a semigroup S is left equalizer simple if and only if the right regular representation of S is a left cancellative semigroup (that is, xa = xb for some x,a,b ∈ S implies a = b). We give a construction, and show that a semigroup is left equalizer simple if and only if it is isomorphic to a semigroup defined by this construction. Let P be a property of semigroups. A congruence ̺ on a semigroup S will be called a P congruence if the factor semigroup S/̺ has the property P. In Section 3, we examine left equalizer simple congruences on semigroups. Let S be a semigroup and ̺ a congruence on S. As in [7], let ̺∗ denote the following congruence on S: (a,b) ∈ ̺∗ for a,b ∈ S if and only if (xa, xb) ∈ ̺ for all x ∈ S. By Corollary 1 of [7], (·)η : ̺ → ̺∗ is a ∧-homomorphism of L(S) into itself. In Theorem 3.1, we show that ̺ is a left equalizer simple congruence on a semigroup S if and only if the congruence ̺∗ is a left cancellative congruence on S. Let S be a semigroup and F a field. For an arbitrary congruence ̺ on S, let F[̺] denote the ideal of the semigroup algebra F[S] which determines the kernel of the extended homomorphism of the semigroup algebra F[S] onto the semigroup algebra F[S/̺] defined by the canonical homomorphism of S onto the factor semigroup S/̺ (see [1]). For an ideal J of F[S], let J|S denote the restriction of the congruence on F[S] defined by J to S. By Lemma 5 of Chapter 4 of [10], for every semigroup S and every field F, the mapping J 7→ J|S is a surjective homomorphism of the ∧-semilattice Id(F[S]) of all ideals of F[S] onto the ∧-semilattice of all congruences on S such that F[̺]|S = ̺ for every congruence ̺ on S. Let (·)κ : ̺ → F[̺]. By the above, κ is an injective mapping of L(S) into Id(F[S]). Using the notation of Section 3.6 of [6] for semigroup algebras, if J is an arbitrary ideal of F[S] then let (J :r F[S]) = {a ∈ F[S]: F[S]a ⊆ J}. It is easy to see that (J :r F[S]) is an ideal of the semigroup algebra F[S] such that J ⊆ (J :r F[S]). An ideal (J :r F[S]) will be called a right colon (more precisely, the right colon of J with respect to F[S]). It is a matter of checking to see that (J1 ∩ J2 :r F[S]) = (J1 :r F[S]) ∩ (J2 :r F[S]) for arbitrary ideals J1 and J2 of F[S]. Thus (·)Φ : J → (J :r F[S]) is a ∧-homomorphism of Id(F[S]) into itself. By the above, we can consider the following diagram. L(S) −→η L(S) κ ↓ ↓ κ (1) −→ Id(F[S]) Φ Id(F(S]) 2 We shall say that the diagram (1) is commutative for some congruence ̺ on a semigroup S if (̺)(η ◦ κ) = (̺)(κ ◦ Φ), that is, ∗ F[̺ ] = (F[̺]:r F[S]). In Theorem 3.2, we prove that, for arbitrary left equalizer simple congruence on a semigroup S, the diagram (1) is commutative. In Section 4, our results will be applied for the matrix representation of finite left equalizer simple semigroups. We also prove a theorem for arbitrary (not necessarily finite) semigroups. We show that if ̺ is a left equalizer simple congruence on a semigroup S then the right colon (F[̺]:r F[S]) equals the augmentation ideal F[ωS] of the semigroup algebra F[S] if and only if the factor semigroup S/̺ is an ideal extension of a left zero semigroup by a null semigroup. 2 Left equalizer simple semigroups Let S be a semigroup and H a non-empty subset of S. By the left equalizer of H we mean the set of all elements x of S for which |xH| = 1, that is, xa = xb is satisfied for all a,b ∈ H. It is clear that the left equalizer of H is either empty or a left ideal of S. Lemma 2.1 On an arbitrary semigroup S, the following assertions are equiv- alent. (i) The left equalizer of any two-element subset of S is either empty or S. (ii) The left equalizer of any subset of S is either empty or S. Proof. Assume (i). Let H be a non-empty subset of S. If |H| = 1 then the left equalizer of H is S. Consider the case when |H| ≥ 2. If x0 ∈ S is in the left equalizer of H then, for every two elements a,b ∈ H with a 6= b, we have x0a = x0b and so x0 belongs to the left equalizer of the subset {a,b}. Thus every x ∈ S is in the left equalizer of {a,b}, that is, xa = xb for all x ∈ S. As a,b ∈ H are arbitrary, we have |xH| = 1 for all x ∈ S and so the left equalizer of H is S. It is obvious that condition (ii) implies condition (i). ⊓ Definition 2.1 A semigroup S will be called a left equalizer simple semigroup if, for arbitrary non-empty subset H of S, the left equalizer of H is either empty or equal to S. Equivalently (see Lemma 2.1), for arbitrary elements a,b ∈ S, the assumption x0a = x0b for some x0 ∈ S implies xa = xb for all x ∈ S. A semigroup S is said to be left simple if S is the only left ideal of S. It is obvious that every left simple semigroup is left equalizer simple. 3 Lemma 2.2 Every M-inversive semigroup is left equalizer simple. Proof. Let x0,a,b be arbitrary elements of an M-inversive semigroup S with x0a = x0b. Then there is an element y ∈ S such that yx0 is a middle unit of S and so, for all x ∈ S, we have xa = xyx0a = xyx0b = xb. Hence S is left equalizer simple. ⊓ The next theorem is an extension of Theorem 1 of [4].
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